Question

Quien puede resolver:

3x²-x, 27x³-1, 9x²-6x+1, 3ax-a+6x-2

16a³x+54x, 12a²x²-42ax²-90x², 32a³x+24a²x-36ax, 32a⁴x-144a²x+162x

5a²+5ax+5ay+5xy, 15a³-15ax²+15a²y-15x²y, 20a³-20ay²+20a²x-20xy², 5a⁵+5a⁴x+5a²y³+5axy³

Answer 1

Tenemos.
Hallar el MCD de las siguientes expresiones
40.
Debes factorizar

3x² - x =       x(3x - 1)

27x³ - 1 =
3³x³ - 1 =        (3x - 1)(9x² + 3x + 1)

9x² - 6x + 1 = (3x - 1)²

3ax - a + 6x - 2 =
(3ax - a) +(6x - 2) =
a(3x - 1) + 2(3x - 1) = (3x - 1)(a + 2)

El MCD es le factor que se repite en las 4 expresiones algebraicas con su menor exponente
(3x - 1)
----------------------------------------------------------------------------------
44.
16a³ + 54x = 2(8a³ - 27)
2(2³a³ + 3³) = 2(2a +3)(4a² - 6a + 9)

12a²x² - 42ax² - 90x² =
6x²( 2a² -  7a - 15) = 6x²(a - 5)(2a + 3)

32a³x + 24a²x - 36ax =
4ax(8a² + 6a - 9) = 4ax(2a + 3)(4a - 3)

32a^4x - 144a²x + 162x =
2x(16a^4 - 72a² + 81) =
2x(4a² - 9)²
El MCD es 2
---------------------------------------------------------------------------------------------
5a² + 5ax + 5ay + 5xy =
(5a² + 5ax) + (5ay + 5xy)
5a(a + x) + 5y(a + x) =
(a + x)(5a + 5y) =
5(a + x)(a + y)

15a³ - 15ax² + 15a²y - 15x²y =
(15a³ + 15a²y) - (15ax² + 15x²y) =
15a²(a + y) - 15x²(a + y) =
(a + y)(15a² - 15x²) =
15(a + y)(a² - x²) =
15(a +y)(a + x)(a - x)

20a³ - 20ay² + 20a²x - 20xy² =
(20a³ - 20ay²) + (20a²x - 20xy²) =
20a(a² - y²) + 20x( a² - y²) =
(a² - y²)(20a + 20x) =
(a² - y²) 20(a + x)
20(a + y)(a - y)(a + x)

5a^5 + 5a^4x + 5a²y³ + 5axy³ =
(5a^5 + 5a^4x) + (5a²y³ + 5axy³) =
5a^4(a + x) + 5ay³(a + x)
(a + x)(5a^4 + 5ay²)
(a + x) 5a(a³ + y³) =
5a(a + x)(a + y)(a² - xy + y²)

El MCD es
5(a + x)(a + y)